> In a more general sense, I have never understood why people insist> on using weak push down autamata (PDA) machine parsing tables when> programming languages provide the full power of Turing Machines.

Because it's notoriously hard to check that grammars encoded as a
program are describing the language they are intended to describe.

Also, PDA grammars can be transformed and subjected to various
interesting algorithms, while no such algorithms exist for
grammars-expressed-as-programs.